Here, we have introduced the notation \(c_\omega :=\cos \omega \)  (we also use \(s_\omega :=\sin \omega \)  later on) and we abuse notation by denoting \(\xi \)  as an element in \(T_xX\)  and \(T^\ast _xX\)  interchangeably. We also have defined \(\xi _v:=\langle \xi ,v\rangle \)  and \(\langle \cdot ,\cdot \rangle \)  denotes the metric inner product in the appropriate sense.
For a proper choice of local coordinates and basis sections, (REF ) coincides with the matrix presentation (149) in [1]} obtained from a five-dimensional setting.
